You will probably get a different answer from everyone, but here is mine: Approach, tool, and process are all pretty generic terms. Approach is perhaps the loosest of all, because it can refer to a methodology, a perspective, and various other things. Tool must be used to refer to something unitary and specific, such as a named test, or a device to do research with, such as a microscope or computer algorithm.
In the social sciences, at least, the others, theory, method, methodology, and framework, have somewhat more specific meanings. A theory is a system of assumptions, principles, and relationships posited to explain a specified set of phenomena. A methodology is often a whole set of methods developed according to a philosophical theory about how best to research and learn about natural or social phenomena.
A framework is a little looser in meaning, and refers to a coherent set of concepts and relationships that are posited about some phenomena. A somewhat more developed and tested framework is a model, and a more fully developed and tested model is a theory!
approach
mental
determine
trajectory
contexts